Suppose the distribution of scores on a college admissions test is mound shaped and
approximately symmetric. The distribution has a mean of 150 and a standard deviation of
25. Use the Empirical Rule on this question and show your work.
(a) Approximately what percentage of students scored between 100 and 175?


To solve this problem using the **Empirical Rule** (also known as the 68-95-99.7 Rule), we need to understand how it applies to a normal distribution. The Empirical Rule states the following for a normal distribution:

1. **68%** of the data falls within **1 standard deviation** of the mean.
2. **95%** of the data falls within **2 standard deviations** of the mean.
3. **99.7%** of the data falls within **3 standard deviations** of the mean.

Given:
- Mean (\(\mu\)) = 150
- Standard deviation (\(\sigma\)) = 25

Let's break down the range of scores step by step:

### Step 1: Define the relevant values

- We want to know the percentage of students who scored between **100** and **175**.
- We will first express these values in terms of how many standard deviations they are from the mean.

1. **100** is how far from the mean:
\[
100 = 150 - 2 \times 25 = 150 - 50
\]
So, 100 is **2 standard deviations below** the mean.

2. **175** is how far from the mean:
\[
175 = 150 + 1 \times 25 = 150 + 25
\]
So, 175 is **1 standard deviation above** the mean.

### Step 2: Apply the Empirical Rule

- From the Empirical Rule:
  - The range between **1 standard deviation below** the mean and **1 standard deviation above** the mean (\( \mu - 1\sigma \) to \( \mu + 1\sigma \)) contains **68%** of the data.
  - The range between **2 standard deviations below** the mean and **2 standard deviations above** the mean (\( \mu - 2\sigma \) to \( \mu + 2\sigma \)) contains **95%** of the data.
  
Now, we're interested in the range between **2 standard deviations below** (100) and **1 standard deviation above** (175). To calculate this:

1. From \( \mu - 2\sigma \) to \( \mu + 2\sigma \), we have **95%** of the data.
2. From \( \mu - 1\sigma \) to \( \mu + 1\sigma \), we have **68%** of the data.
3. The percentage of data between \( \mu - 2\sigma \) (100) and \( \mu - 1\sigma \) (125) can be calculated as:
\[
\frac{95\% - 68\%}{2} = 13.5\%
\]

Thus, the percentage of students scoring between **100** and **175** is the sum of the following:
- The percentage between \( \mu - 2\sigma \) (100) and \( \mu - 1\sigma \) (125): **13.5%**
- The percentage between \( \mu - 1\sigma \) (125) and \( \mu + 1\sigma \) (175): **68%**

\[
13.5\% + 68\% = 81.5\%
\]

### Final Answer:
Approximately **81.5%** of students scored between 100 and 175.

Would you like further clarification or additional details?

Here are some related questions you might want to explore:
1. What is the probability that a student scores between 125 and 150?
2. What percentage of students scored above 175?
3. How many standard deviations above the mean is a score of 200?
4. What is the probability that a student scores between 150 and 200?
5. If the test scores are approximately normal, how would you find the probability of a student scoring below 100?

**Tip**: The Empirical Rule is a handy tool for quick estimates, but for exact probabilities, using the Z-score and standard normal distribution tables would give more precise results.

Suppose the distribution of scores on a college admissions test is mound shaped and approximately symmetric. The distribution has a mean of 150 and a standard deviation of 25. Use the Empirical Rule on this question and show your work. (a) Approximately what percentage of students scored between 100 and 175?

Learn how to use the Empirical Rule to estimate the percentage of students who scored between 100 and 175 on a college admissions test. This problem involves a normal distribution with a mean of 150 and a standard deviation of 25. Suitable for high school students in Grades 10-12.

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